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Form 4 Maths
Chapter 1 - Number Systems * Weak concept about what is a real number / purely imaginary number : if a+bi (a'', ''b are real) is a real number, then b = 0 : if it is a purely imaginary number, then a = 0 : if it is a non-real number, then b \ne 0 * Note that a'' is the real part and ''b is the imaginary part * They do not know if z_{1}, z_{2} are complex numbers and z_{1} x = z_{2} , then x = \frac{z_{2}}{z_{1}} . Instead, they will expand everything and compare the real parts and imaginary parts. Chapter 2 - Quadratic Equations * Not using the fastest method. e.g. solve 7(x+4)^2 + 30 = 2 by expanding everything and then use the quadratic formula. * Some students are weak in simplifying surd, especially when it involves a non-real number. * When asked to form a quadratic equation with two given roots, they usually omitted either the middle x'' or the '=0' in the final answer. * Weak in question involving many unknowns, e.g. nature of roots of x^2 + ax + b = 0 if a>0 and b < 0 Chapter 3 - Functions and Graphs (I) * Generally very weak in reading graph ** E.g. after getting the ''y-''coordinate of the vertex from the graph, instead of reading the ''x-coordinate directly, they sub. the value of y'' into the function to solve for the value of ''x * They confused when one function y = f(x) but they are asked to use the graph of y = g(x) . * Forget to label the line draw on the coordinate plane. * The effect of different a'' to the width of opening of the graph of y = ax^2 + bx + c Chapter 4 - Functions and Graphs (II) * Weak concept about function * Weak and even dislike graphs * Weak in understanding (or even just remembering) different types of transformation ** Especially when the function is given in the tabular form. * As students usually just 'remember' the rule, they confused when they are asked about the new location of a certain point as well as the analytic form of the new function under transformation * Enlargement and reduction are generally difficult to the students, esp. when it is done along the ''x-axis Chapter 5 - Exponential and Logarithmic Functions * Weak in solving both exponential equations and logarithmic equations. * The situation gets worse when these equations are reducible to quadratic equation (chapter 12 - more about equations (II)) * change the base of log to 10 all the time, even when it is not necessary * Make up new rules (e.g. distributive law) for logarithm * For exponential equations, they take log when it is not necessary (or even not appropriate), then they repeat the mistake in the previous bullet point. * Don't have the habit to verify the final answer. Hence, usually forgot to reject the extra roots. ** After falling into the trap for many times, some always reject the negative root(s), even in solving other types of equations. * Cannot solve equation like k \log (a-9) = 0 ** It may reflect the fact that they usually do not use factorization to solve equation. ** Similar situation happens when they are asked to simplify certain expressions, they prefer expanding everything instead of factorizing the expression Chapter 6 - Equations of Straight Lines * Weak in understanding that if a point (a, b) lies on the straight line y = mx + c , then the coordinates can be expressed as (a, ma+c) * Weak in finding the coordinates of circumcenter and orthocenter (esp. in obtuse-angled triangle). * Weak in remembering/understanding the equations of vertical/horizontal lines * Weak in handling rotation of straight lines (even in the simplest case that a straight line ( y=mx ) passes through the origin is rotated 90^\circ about the origin. ** Even worse if the resulting straight line becomes vertical/horizontal. Chapter 7, 8 - Circle Geometry * They do not know that the name of the polygon, e.g. ABCD, reflect a clockwise/anti-clockwise order on the points A, B, C and D. * Often omit the reason: corr. sides, \sim \Delta s * Weak in using the exterior angle-related theorems ** They even think that there is only one exterior angle * Weak in handling reflex angle Chapter 11 - More about Equations (I) Chapter 12 - More about Equations (II) * Very weak in solve application problem. E.g.: ** The daily sales amount of good A will decrease by 10kg if the price per kg of the good is increased by $1. When the price is $30 per kg, the sales amount is 300 kg. Find a relation between the sales amount and the price x.